A vast majority of GIS digital maps represent straight linear features—streets, water, political, land use, or recreational boundaries—as chained line segments connected by mutual endpoints often referred to as shape points. These points represent a point in 2-space or 3-space deemed to be along the path of the line of travel, service, or boundary. These shape points are usually inflections or bends along a single path, such that when two line edges meet at any shape point they are generally not collinear. The clear intention of such lines is representing the real-world feature to a reasonably good approximation in a reasonably simple and compact form. Travel along the actual real-world path or boundary can be approximately represented by traversing each successive line segment.
The shaped line format described is a simplified model of real-world paths; its primary strength is its simplicity. It is easy to draw on a raster screen, and it readily allows computations for length, distance, and other geometric queries. However, since actual linear features are much more complex and composed of non-line segment components, representation by shaped lines is plagued with representation error. Any attempt to reduce this representation error requires an increase in the density of the data; no finite amount of data stored in a line segment format can perfectly represent a non-line segment shape.
The Clothoid is a two-dimensional shape or path, defined to have constant change in curvature over the travel distance. As classically illustrated, the clothoid looks like a clock spring, with zero curvature at one end, and then coiling ever tighter at the other end. Arcs and line segments are simply special cases of the clothoid. A circular arc is a clothoid because it has a constant curvature—that is, zero curvature change—wherein the curvature magnitude is inversely related to that circle's radius. A straight line is also a clothoid, having both a no-curvature change, and a constant curvature of zero over its entire length.
Clothoids and their special cases, circular arcs and straight lines, are used in much real-world construction. Roads in particular are often constructed from pieced segments consisting of straight lines, circular arcs, and clothoids. Roadbed designers recognize that roadway curvature directly relates to movement of steering wheels and axle components of vehicles traversing the roadway. For them, limiting any abrupt changes in curvature by choosing clothoid design where feasible, represents a decision that maximizes vehicular safety and comfort, and minimizes wear on roadway components.
The concept of splines is available in mathematics. Though originally referring to a thin flexible rod used to draw curves, the term is mathematically understood as a function fit in which the fitting function has some number of continuous derivatives. Thinking further about the vehicle and steering wheel example above, one can see that in order to minimize disruptive changes in steering, a vehicular path should be a spline with respect to heading change over distance traveled, with the first derivative of heading change per unit distance—that is, curvature—being a continuous function. This type of function will be referred to herein as the “clothoid spline.”